1. The derivative of the function f(x)=x3−6x2+4x−7f(x) = x^3 - 6x^2 + 4x - 7f(x)=x3−6x2+4x−7
is:
A. 3x2−12x+43x^2 - 12x + 43x2−12x+4
B. 3x2−12x−43x^2 - 12x - 43x2−12x−4
C. 3x2−6x+43x^2 - 6x + 43x2−6x+4
D. 3x2−6x−43x^2 - 6x - 43x2−6x−4
Answer: a) 3x2−12x+43x^2 - 12x + 43x2−12x+4
Rationale: The derivative of f(x)=x3−6x2+4x−7f(x) = x^3 - 6x^2 + 4x - 7f(x)=x3−6x2+4x−7 is
found by applying the power rule:
f′(x)=3x2−12x+4f'(x) = 3x^2 - 12x + 4f′(x)=3x2−12x+4.
2. Solve 3x+2=03x + 2 = 03x+2=0:
A. x=−23x = -\frac{2}{3}x=−32
B. x=23x = \frac{2}{3}x=32
C. x=−2x = -2x=−2
D. x=2x = 2x=2
Answer: a) x=−23x = -\frac{2}{3}x=−32
Rationale: Solve 3x+2=03x + 2 = 03x+2=0 by isolating xxx:
3x=−23x = -23x=−2
x=−23x = -\frac{2}{3}x=−32.
3. The second derivative of f(x)=4x4−3x3+2x2f(x) = 4x^4 - 3x^3 + 2x^2f(x)=4x4−3x3+2x2 is:
A. 48x2−18x+448x^2 - 18x + 448x2−18x+4
B. 48x2−18x48x^2 - 18x48x2−18x
C. 48x2−12x48x^2 - 12x48x2−12x
D. 48x2−6x48x^2 - 6x48x2−6x
,Answer: a) 48x2−18x+448x^2 - 18x + 448x2−18x+4
Rationale:
First derivative: f′(x)=16x3−9x2+4xf'(x) = 16x^3 - 9x^2 + 4xf′(x)=16x3−9x2+4x
Second derivative: f′′(x)=48x2−18x+4f''(x) = 48x^2 - 18x + 4f′′(x)=48x2−18x+4.
4. The sum of the roots of the quadratic equation 2x2+5x−3=02x^2 + 5x - 3 = 02x2+5x−3=0 is:
A. −52-\frac{5}{2}−25
B. 52\frac{5}{2}25
C. −53-\frac{5}{3}−35
D. 53\frac{5}{3}35
Answer: a) −52-\frac{5}{2}−25
Rationale: Using the sum of roots formula −ba\frac{-b}{a}a−b for ax2+bx+c=0ax^2 + bx + c =
0ax2+bx+c=0, we get:
Sum = −52\frac{-5}{2}2−5.
5. The value of ∫3x2 dx\int 3x^2 \, dx∫3x2dx is:
A. x3+Cx^3 + Cx3+C
B. x3−Cx^3 - Cx3−C
C. x3+Cx^3 + Cx3+C
D. x3x^3x3
Answer: a) x3+Cx^3 + Cx3+C
Rationale:
The integral of 3x23x^23x2 is x3+Cx^3 + Cx3+C.
6. Find the derivative of y=ln(x2+1)y = \ln(x^2 + 1)y=ln(x2+1):
A. 2xx2+1\frac{2x}{x^2 + 1}x2+12x
B. 1x2+1\frac{1}{x^2 + 1}x2+11
, C. xx2+1\frac{x}{x^2 + 1}x2+1x
D. 2xx+1\frac{2x}{x + 1}x+12x
Answer: a) 2xx2+1\frac{2x}{x^2 + 1}x2+12x
Rationale:
Using the chain rule for ln(u)\ln(u)ln(u), where u=x2+1u = x^2 + 1u=x2+1, we get the
derivative as 2xx2+1\frac{2x}{x^2 + 1}x2+12x.
7. The roots of the quadratic equation x2−4x+4=0x^2 - 4x + 4 = 0x2−4x+4=0 are:
A. 222
B. −2-2−2
C. 444
D. 000
Answer: a) 222
Rationale: The equation factors as (x−2)2=0(x - 2)^2 = 0(x−2)2=0, so the root is x=2x = 2x=2.
8. Find the value of ∫13(2x+1) dx\int_1^3 (2x + 1) \, dx∫13(2x+1)dx:
A. 14
B. 15
C. 13
D. 12
Answer: b) 15
Rationale:
∫13(2x+1) dx=[x2+x]13=(9+3)−(1+1)=15\int_1^3 (2x + 1) \, dx = \left[ x^2 + x \right]_1^3 = (9
+ 3) - (1 + 1) = 15∫13(2x+1)dx=[x2+x]13=(9+3)−(1+1)=15.
9. The derivative of f(x)=sin(2x)f(x) = \sin(2x)f(x)=sin(2x) is:
A. cos(2x)\cos(2x)cos(2x)
B. 2cos(2x)2\cos(2x)2cos(2x)
is:
A. 3x2−12x+43x^2 - 12x + 43x2−12x+4
B. 3x2−12x−43x^2 - 12x - 43x2−12x−4
C. 3x2−6x+43x^2 - 6x + 43x2−6x+4
D. 3x2−6x−43x^2 - 6x - 43x2−6x−4
Answer: a) 3x2−12x+43x^2 - 12x + 43x2−12x+4
Rationale: The derivative of f(x)=x3−6x2+4x−7f(x) = x^3 - 6x^2 + 4x - 7f(x)=x3−6x2+4x−7 is
found by applying the power rule:
f′(x)=3x2−12x+4f'(x) = 3x^2 - 12x + 4f′(x)=3x2−12x+4.
2. Solve 3x+2=03x + 2 = 03x+2=0:
A. x=−23x = -\frac{2}{3}x=−32
B. x=23x = \frac{2}{3}x=32
C. x=−2x = -2x=−2
D. x=2x = 2x=2
Answer: a) x=−23x = -\frac{2}{3}x=−32
Rationale: Solve 3x+2=03x + 2 = 03x+2=0 by isolating xxx:
3x=−23x = -23x=−2
x=−23x = -\frac{2}{3}x=−32.
3. The second derivative of f(x)=4x4−3x3+2x2f(x) = 4x^4 - 3x^3 + 2x^2f(x)=4x4−3x3+2x2 is:
A. 48x2−18x+448x^2 - 18x + 448x2−18x+4
B. 48x2−18x48x^2 - 18x48x2−18x
C. 48x2−12x48x^2 - 12x48x2−12x
D. 48x2−6x48x^2 - 6x48x2−6x
,Answer: a) 48x2−18x+448x^2 - 18x + 448x2−18x+4
Rationale:
First derivative: f′(x)=16x3−9x2+4xf'(x) = 16x^3 - 9x^2 + 4xf′(x)=16x3−9x2+4x
Second derivative: f′′(x)=48x2−18x+4f''(x) = 48x^2 - 18x + 4f′′(x)=48x2−18x+4.
4. The sum of the roots of the quadratic equation 2x2+5x−3=02x^2 + 5x - 3 = 02x2+5x−3=0 is:
A. −52-\frac{5}{2}−25
B. 52\frac{5}{2}25
C. −53-\frac{5}{3}−35
D. 53\frac{5}{3}35
Answer: a) −52-\frac{5}{2}−25
Rationale: Using the sum of roots formula −ba\frac{-b}{a}a−b for ax2+bx+c=0ax^2 + bx + c =
0ax2+bx+c=0, we get:
Sum = −52\frac{-5}{2}2−5.
5. The value of ∫3x2 dx\int 3x^2 \, dx∫3x2dx is:
A. x3+Cx^3 + Cx3+C
B. x3−Cx^3 - Cx3−C
C. x3+Cx^3 + Cx3+C
D. x3x^3x3
Answer: a) x3+Cx^3 + Cx3+C
Rationale:
The integral of 3x23x^23x2 is x3+Cx^3 + Cx3+C.
6. Find the derivative of y=ln(x2+1)y = \ln(x^2 + 1)y=ln(x2+1):
A. 2xx2+1\frac{2x}{x^2 + 1}x2+12x
B. 1x2+1\frac{1}{x^2 + 1}x2+11
, C. xx2+1\frac{x}{x^2 + 1}x2+1x
D. 2xx+1\frac{2x}{x + 1}x+12x
Answer: a) 2xx2+1\frac{2x}{x^2 + 1}x2+12x
Rationale:
Using the chain rule for ln(u)\ln(u)ln(u), where u=x2+1u = x^2 + 1u=x2+1, we get the
derivative as 2xx2+1\frac{2x}{x^2 + 1}x2+12x.
7. The roots of the quadratic equation x2−4x+4=0x^2 - 4x + 4 = 0x2−4x+4=0 are:
A. 222
B. −2-2−2
C. 444
D. 000
Answer: a) 222
Rationale: The equation factors as (x−2)2=0(x - 2)^2 = 0(x−2)2=0, so the root is x=2x = 2x=2.
8. Find the value of ∫13(2x+1) dx\int_1^3 (2x + 1) \, dx∫13(2x+1)dx:
A. 14
B. 15
C. 13
D. 12
Answer: b) 15
Rationale:
∫13(2x+1) dx=[x2+x]13=(9+3)−(1+1)=15\int_1^3 (2x + 1) \, dx = \left[ x^2 + x \right]_1^3 = (9
+ 3) - (1 + 1) = 15∫13(2x+1)dx=[x2+x]13=(9+3)−(1+1)=15.
9. The derivative of f(x)=sin(2x)f(x) = \sin(2x)f(x)=sin(2x) is:
A. cos(2x)\cos(2x)cos(2x)
B. 2cos(2x)2\cos(2x)2cos(2x)
